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Differential Equations

Short Answer Type

101. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

straight e to the power of straight y dx plus straight e to the power of straight x dy space equals space 0

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102. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

left parenthesis sinx plus cosx right parenthesis space dy space equals space left parenthesis cosx minus sinx right parenthesis space dx

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103. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

straight y apostrophe space equals left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis

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104. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
$left parenthesis straight e to the power of straight x plus straight e to the power of straight x right parenthesis space dy space minus space left parenthesis straight e to the power of straight x minus straight e to the power of straight x right parenthesis space dx space equals space 0$

The given differential equation is

left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis space dy space equals space left parenthesis straight e to the power of straight x minus straight e to the power of negative straight x end exponent right parenthesis space dx

therefore space space space space space space space space space space space space space space space space space dy space equals space open parentheses fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction close parentheses dx

Integrating,

integral dy space equals space integral fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction dx

therefore space space space space space space space space space space space space space space space space space straight y space equals space log space left parenthesis straight e to the power of straight x plus straight e to the power of negative straight x end exponent right parenthesis space plus straight c comma space space space space straight x space element of space straight R space is space the space solution. space

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105.

Solve the differential equation
$left parenthesis tan squared straight x plus 2 space tanx space plus 5 right parenthesis space dy over dx space equals space 2 left parenthesis 1 plus tanx right parenthesis space sec squared straight x.$